The prompt presents two assertions concerning the chemistry of manganese compounds and requires their validity assessment through chemical principles and reactions.
Statement I: The fusion of \(MnO_2\) with \(KOH\) and an oxidizing agent yields the dark green compound \(K_2MnO_4\).
Explanation: When manganese dioxide (\(MnO_2\)) is heated with potassium hydroxide (\(KOH\)) in the presence of an oxidizing agent like chlorine (\(Cl_2\)) or potassium nitrate (\(KNO_3\)), potassium manganate (\(K_2MnO_4\)), which is dark green, is formed.
The reaction is: \(2MnO_2 + 4KOH + O_2 \rightarrow 2K_2MnO_4 + 2H_2O\)
Consequently, Statement I is confirmed as true.
Statement II: Electrolytic oxidation of the manganate ion in an alkaline medium produces the permanganate ion.
Explanation: Within an alkaline environment, the manganate ion (\(MnO_4^{2-}\)) can undergo further oxidation to yield the permanganate ion (\(MnO_4^{-}\)), particularly under electrolytic conditions. This transformation results from the \(MnO_4^{2-}\) ion gaining an electron:
\(MnO_4^{2-} \ + \ e^- + \ H_2O \rightarrow \ MnO_4^- + 2OH^-\)
This indicates an increase in the oxidation state of manganese from +6 in manganate to +7 in permanganate.
Therefore, Statement II is also established as true.
Conclusion: Both Statement I and Statement II are validated by the provided explanations. The correct determination is: Both Statement I and Statement II is true.